A206290 - OEIS

%I #17 Nov 06 2014 08:35:13

%S 1,1,2,3,5,7,12,17,29,44,77,114,218,330,617,987,1913,2968,6068,9500,

%T 19263,31399,64268,101702,218891,348559,735823,1205239,2576727,

%U 4119884,9100854,14588992,31841260,52163378,114485092,183947681,414704366,667453931,1487920000

%N G.f.: Sum_{n>=0} Product_{k=1..n} Series_Reversion( x/(1 + x^k) ).

%C Compare to the g.f. of partition numbers (A000041): Sum_{n>=0} Product_{k=1..n} x/(1 - x^k).

%H Vaclav Kotesovec, <a href="/A206290/b206290.txt">Table of n, a(n) for n = 0..250</a>

%F G.f.: Sum_{n>=0} Product_{k=1..n} G_k(x), where G_n(x) is defined by:

%F (1) G_n(x) = Series_Reversion( x/(1 + x^n) ),

%F (2) G_n(x) = x + x*G_n(x)^n,

%F (3) G_n(x) = Sum_{k>=0} binomial(n*k+1, k) * x^(n*k+1) / (n*k+1).

%e G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 12*x^6 + 17*x^7 +...

%e such that, by definition,

%e A(x) = 1 + G_1(x) + G_1(x)*G_2(x) + G_1(x)*G_2(x)*G_3(x) + G_1(x)*G_2(x)*G_3(x)*G_4(x) +...

%e where G_n( x/(1 + x^n) ) = x.

%e The first few expansions of G_n(x) begin:

%e G_1(x) = x + x^2 + x^3 + x^4 + x^5 + x^6 +...+ x^(n+1) +...

%e G_2(x) = x + x^3 + 2*x^5 + 5*x^7 + 14*x^9 +...+ A000108(n)*x^(2*n+1) +...

%e G_3(x) = x + x^4 + 3*x^7 + 12*x^10 + 55*x^13 +...+ A001764(n)*x^(3*n+1) +...

%e G_4(x) = x + x^5 + 4*x^9 + 22*x^13 + 140*x^17 +...+ A002293(n)*x^(4*n+1) +...

%e G_5(x) = x + x^6 + 5*x^11 + 35*x^16 + 285*x^21 +...+ A002294(n)*x^(5*n+1) +...

%e G_6(x) = x + x^7 + 6*x^13 + 51*x^19 + 506*x^25 +...+ A002295(n)*x^(6*n+1) +...

%e G_7(x) = x + x^8 + 7*x^15 + 70*x^22 + 819*x^29 +...+ A002296(n)*x^(7*n+1) +...

%e Note that G_n(x) = x + x*G_n(x)^n.

%o (PARI) {a(n)=polcoeff(sum(m=0,n,prod(k=1,m,serreverse(x/(1+x^k+x*O(x^n))))),n)}

%o for(n=0,45,print1(a(n),", "))

%Y Cf. A206289, A194560, A110448.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 05 2012